MATH 166, B1,B2,C1,C2,D1,D2
Workshop on Calculus I material, Tue. Aug. 27
Get into groups of 2 or 3. Do as many problems as you can. No calculators are permitted.
When the TA calls time, submit one paper with your answers and the names of the members
of your team (along with your section number). Each member of the team with most number
of points will get a guaranteed 10/10 on Quiz 1.
1
9am Version
1. Find the indicated limit or determine that it does not exist.
sin θ
(a) (1 point) lim
θ→0 tan 3θ
Solution:
1
3
x2 + x − 6
x→2
x−2
(b) (1 point) lim
Solution: 5
x2 − 4x + 6
x→3
x2 − 9
(c) (1 point) lim
Solution: Does Not Exist
(d) (1 point) lim
x→1
x−1
x − x3
Solution: −
(e) (1 point) x→∞
lim
1
2
5x − 1
3x2 + 14
Solution: 0
2. Evaluate the following derivatives. Simplify.
i
d h 2
(a) (1 point)
(3x + x + 1)2010
dx
Solution: 2010(3x2 + x + 1)2009 (6x + 1)
(b) (1 point)
d sin2 x
dx cos x
Solution: sin x + tan x sec x
(c) (1 point) Find Dx y for y = sec2 (4x).
Solution: 8 tan(4x) sec2 (4x)
(d) (1 point) Find f 0 (3) for f (x) = 2 + x2
4/3
√
3
Solution: 8 11
(e) (1 point) Find
dy
using implicit differentiation for the curve 2x − y 2 = cos(xy) + 5.
dx
2 + y sin(xy)
2y − x sin(xy)
Solution:
3. (1 point) Find Dx101 cos x.
Solution: − sin x
4. Find the following definite integrals.
Z 3
5x + 5
dx
(a) (1 point)
2
1 x + 2x + 2
5 17
ln
2
5
Solution:
(b) (1 point)
Z 1
x cos x3 dx
−1
Solution: 0
5. Find the following indefinite integrals.
(a) (1 point)
Z
x exp −x2 dx
1
2
Solution: − e−x + C
2
Page 2
(b) (1 point)
Z
1 + 4 ln x
dx
x
Solution: 2 ln2 x + ln x + C
6. Find the derivatives of the following functions.
(a) (1 point) h(x) = x3 3x
Solution: 3x
2 −1
2 −1
x2 x2 ln 9 + 3
√
x x−3
(b) (1 point) y = ln
(4x + 5)10
Solution:
68x2 − 231x + 30
−8x3 + 14x2 + 30x
7. (1 point) Find the derivative of the function y(x) given for −3 < x < −1 by 0 < y < π
and cos y = x + 2.
Solution: − q
1
1 − (x + 2)2
dy
√
8. (1 point) Find that solution of the differential equation
= x3 y which satisfies y = 2
dx
when x = 0.
!2
Solution: y(x) =
x4 √
+ 2
8
Page 3
2
10am Version
1. Evaluate each limit, or determine that it does not exist.
x2 − 9
(a) (1 point) lim
x→−3 x + 3
Solution: −6
sin x tan x
x→0
x2
(b) (1 point) lim
Solution: 1
x2 − 3x
x2 − 4x + 3
(c) (1 point) lim+
x→1
Solution: +∞ or Does Not Exist
v
u
u
t
(d) (1 point) x→∞
lim
4 − x2
(−x + 2)(x + 2)
Solution: 1
2. Evaluate Dx y:
(a) (1 point) y = 5 − x2 sec x
Solution: −x2 tan x sec x + 5 tan x sec x − 2x sec x
(b) (1 point) y =
Solution:
tan 4x
4x
8x sec2 (4x) − 2 tan(4x)
8x2
(c) (1 point) y =
1
.
(3x + 1)2
Solution: −
6
(3x + 1)3
(d) (1 point) y = sin3 x − 4 cos2 x
Solution: 3 sin2 x cos x + 8 sin x cos x
Page 4
(e) (1 point) xy 2 + x2 y − 2 = 0.
−y 2 − 2xy
x2 + 2xy
Solution:
3. (1 point) Find y 00 if y =
3
.
sin 3x
Solution: 27 csc3 (3x) + 27 csc(3x) cot2 (3x)
4. Find
dy
.
dx
√
(a) (1 point) y =
√
Solution:
√
e2x + e
2x
√
e2x
e 2x
+√
2x
(b) (1 point) y(x) = ln (x − 1)5 (x2 + 2)2
9x2 − 4x + 10
(x − 1)(x2 + 1)
Solution:
(c) (1 point) y = xsin x
Solution:
sin x
+ ln x cos x xsin x
x
(d) (1 point) y = sin−1 (3x)
Solution: √
3
1 − 9x2
5. Evaluate the definite or indefinite integral.
Z 1 √ (a) (1 point)
x3 + 3 x dx
−1
Solution: 0
Page 5
(b) (1 point)
Z
34x−1 dx
34x−1
+C
4 ln 3
Solution:
(c) (1 point)
Z
sin x
dx
1 + cos x
Solution: − ln(1 + cos x) + C
(d) (1 point)
Z
sin x (1 + 2 cos x)100 dx
Solution: −
(e) (1 point)
Z 7
1
1
(1 + 2 cos x)101 + C
202
1
dx
x+3
Solution: ln
(f) (1 point)
Z 1
√
0
Solution:
5
2
1
dx
4 − x2
π
6
Page 6
3
8am Version
1. Find the indicated limit or determine that it does not exist.
3n2 − 4n + 8
(a) (1 point) lim
n→∞ 5n2 + 6n − 1
Solution:
3
5
u−2
u→2 u2 − 4
(b) (1 point) lim
Solution:
1
4
sin(5θ) cos(3θ)
θ→0
sin(2θ)
(c) (1 point) lim
Solution:
5
2
(d) (1 point) lim−
x→2
x2
(x − 2)(3 − x)
Solution: −∞ or Does Not Exist
dy
for each of the following:
dx
(a) (1 point) y = 2πx3 + 2x + 5x−2 − 3x2/3
2. Find
Solution: 6πx2 + 2 − 10x−3 − 2x−1/3
(b) (1 point) y = sin3 x2 + 1
Solution: 6x sin2 (x2 + 1) cos(x2 + 1)
(c) (1 point) y = x2
Solution:
√
3x + 1
√
3x2
√
+ 2x 3x + 1
2 3x + 1
Page 7
(d) (1 point) y =
Solution:
3. (1 point) Find
Solution:
3x
(x2 + 1)3
3(x2 + 1)3 − 12x2 (x2 + 1)2
(x2 + 1)6
dy
if x3 y 2 = 7 − y + x2 .
dx
2x − 3x2 y 2
1 + 2x3 y
4. (1 point) If f (x) = 2x cos(x), find f 00 (x).
Solution: −2x cos(x) − 4 sin x
Page 8
5. Evaluate the following derivatives:
i
d h √ 4
(a) (1 point)
ln x + 1
dx
Solution:
(b) (1 point)
d
[arctan(2x + 1)]
dx
Solution:
(c) (1 point)
2x3
x4 + 1
2
1 + (2x + 1)2
d h x√ 3 i
7
dx
√
Solution: 7
3x
√
(ln 7) 3
d Z 1 t+1
(d) (1 point)
dt
dx x t4 + 1
Solution: −
x+1
x4 + 1
x3 d 2
6. (1 point) Find
x +1
.
dx
Solution: x2 + 1
x3
"
2x4
+ 3x2 ln(x2 + 1)
x2 + 1
7. (1 point) Simplify the following: cos sin−1 x .
Solution:
√
1 − x2
8. Evaluate the following integrals:
(a) (1 point)
Z 2 6x3 + 3x2 − 3x dx
−2
Solution: 16
Page 9
#
(b) (1 point)
Z 3 sin 2x + 4e3x dx
4
3
Solution: − cos(2x) + e3x + C
2
3
9. Evaluate the following integrals:
(a) (1 point)
Z
e3 ln x dx
1 4
x +C
4
Solution:
(b) (1 point)
Z
ex
dx
ex + 2
Solution: ln(ex + 2) + C
(c) (1 point)
Z 2
√
0
t3
dx (Simplify your answer.)
t4 + 9
Solution: 1
(d) (1 point)
Z 1
√
0
Solution:
1
dx (Simplify your answer.)
4 − x2
π
6
Page 10